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A monolayer of carbon is called graphene. It exhibits unusual properties in the Hall effect and in the cyclotron resonance. It is found that it exhibits fractional charge in the Hall effect. The interactions amongst electrons almost become constant at low temperatures. Hence, the Kohn's theorem, which shows that the interactions do not play much role in determining the cyclotron resonance, becomes operative at low temperatures. The experiments on graphene do not depend on the wave vector dependence of the frequency. Hence whether the dispersion depends on k2 or on k does not matter. The Hubbard model has been very successful in explaining the ground state of several electron systems. We consider a triangle with three vortices. Each vortex can be occupied by two electrons. By using the spin in a particular way, we can obtain new features in the Hubbard model. There is a doubling in the Peierls-Luttinger phase factor and eigen values acquire higher multiplicities than are known for the usual treatment of spin. The flux is distributed on the area of the triangle. The graphene consists of hexagons of carbon atoms but the Hall effect shows that there are defects on which electrons form clusters so that there is spin wave type behaviour. A cluster of electrons shows spin-waves leading to "spin deviation" of several per cent.

We make clusters of atoms of the size of less than 1 nanometer by using the density functional theory and from that we obtain the bond lengths corresponding to the minimum energy configuration. We are able to optimize large clusters of atoms and find the vibrational frequencies for each cluster. This calculation provides us with a method to identify the clusters present in an unknown sample of a glass by comparing the experimental Raman frequency with the calculated value. We start with the experimental values of the Raman frequencies of PSe (Phosphorous-Selenium) glass. We calculate the structural parameters of PSe, P4Se, P2Se2, P4Se5, PSe4, P4Se3 clusters of atoms and tabulate the vibrational frequencies. We compare the calculated values with those measured. In this way we find the clusters of atoms present in the glass. Some times, the same number of atoms can be rearranged in a different symmetry. Hence we learn the symmetries of molecules. We find that certain symmetries are broken due to self-organization in the glassy state.

The 30 nm wide quantum wells on a 4x4 mm2 piece of GaAs/AlGaAs are formed when the layers of GaAs are deposited on AlGaAs films. The two-dimensional density of electrons is 3x1011 cm-2 and the mobility is 32x106 cm2/Vs. In such a sample the Hall resistivity as a function of magnetic field is not a linear function. Hence a suitable theory to understand the Hall effect is formulated. We find that there are phase transitions as a function of temperature. There are lots of fractions of charge which are explained on the basis of spin and orbital angular momentum of the electron. The nano meter size films of graphite also show that the Hall resistivity is non-linear and shows steps as a function of magnetic field. We make an effort to understand the steps in the Hall effect resistivity of graphite with quantum wells formed on the surface. It is found that the fractions are in four categories, (i) the principal fractions which are determined by spin and orbital angular momenta, (ii) the resonances which occur at the difference between two values such as =1-2, (iii) two-particle states which occur at the sum of the two frequencies and (iv) there are clusters of electrons localized in some areas of the sample. The spin in the clusters is polarized so that it becomes NS which is not 1/2 but depends on the number N, of electrons in a cluster.

We have calculated the vibrational frequencies of clusters of Ga and As atoms from the first principles using the density-functional theory (DFT) method and the local-density approximation (LDA). We find that the calculated value of 286.2 cm(-1) for a linear cluster of Ga(2)As(2) is very near the experimental value of 292 ± 4 cm(-1). The calculated value of 289.4 cm(-1) for Ga(2)As(6) (dumb bell) cluster is indeed very near the experimental value. There are strong phonon correlations so that the cluster frequency is within the dispersion relation of the crystal LO value. There is a weak line in the experimental Raman spectrum at 268 cm(-1) which is very near the value of 267.3 cm(-1) calculated for the Ga(2)As (triangular) cluster. The weak lines corresponding to the linear bonds provide the strength to the amorphous samples. There are clusters of atoms in the glassy state of GaAs.

We have used the density functional theory to make a toxicity prediction model of water-soluble arsenicals (WSA). The structures have been optimised for the minimum energy of the Schrodinger equation. In the present work, the usefulness of electrophilicity and charge transfer in predicting the toxicity of WSA, namely, monomethylarsenic acid (MMA) (III), dimethylarsenic acid (DMA) (III), arsenic acid, arsenous acid, MMA (V) and DMA (V), is assessed. It is demonstrated that the toxicity of arsenicals (both electron donors and acceptors) in gas and solution phases can be adequately explained in terms of electrophilicities. Amount of charge transfer between the WSA and the biosystem, simulated as nucleic acid (NA) bases and DNA base pairs, indicates the importance of charge transfer in experimental toxicity. Interaction of arsenicals with NA bases/selected base pairs is determined using Paar's charge-transfer formula. The experimental toxicity (LD50) of arsenicals are taken as dependent variables and the ener...

The Hall resistivity in the layers of AlxGa1-xAs/Al0.32Ga0.68As is found to show plateaus at certain fractions which depend on the effective charge. The Hall resistivity formula ρxy=h/e2 has been modified to ρxy=h/[(1/2) ge2] so that the effective charge of the electron becomes, e*=(1/2) ge. The plateaus occur at the effective charge determined by g = (2j+1)/(2l+1). Some of the plateaus are explained to arise from the g values while some others require the use of Landau levels. The flux quantization is modified to include the effect of spin. When the samples are doped with aluminium, the clusters of Al atoms occur in the GaAs resulting into electron clusters in which the spin is NS with S=1/2 and N=101. The electron clusters form a temperature dependent plateau in the Hall resistivity.

We have calculated the vibrational frequencies of clusters of atoms from the first principles by using the density-functional theory in the local density approximation (LDA). We are also able to calculate the electronic binding energy for all of the clusters of atoms from the optimized structure. We have made clusters of BanOm (n, m=1-6) and have determined the bond lengths, vibrational frequencies as well as intensities in each case. We find that the peroxide cluster BaO2 occurs with the O-O vibrational frequency of 836.3 cm(-1). We also find that a glass network occurs in the material which explains the vibrational frequency of 67 cm(-1). The calculated values agree with those measured from the Raman spectra of barium peroxide and Ba-B-oxide glass. We have calculated the vibrational frequencies of BaO4, GeO4 and SiO4 each in tetrahedral configuration and find that the vibrational frequencies in these systems depend on the inverse square root of the atomic mass.

In 1983, Laughlin reported a wave function which while using the first-principles kinetic energy and Coulomb interactions fractionalizes the charge of the electron so that a charge such as 1/3 occurs. Since then this wave function has been applied to many problems in condensed matter physics. An effort is made to review the literature dealing with Aharonov–Bohm effect, ground state, confinement, phase transitions, Wigner and Luttinger solids, edge states, Anderson's theory, statistics and anyons, etc. The importance of the angular momentum is pointed out and it is shown that Landau levels play an important role in understanding the fractions at which the plateaus occur in the quantum Hall effect.

We have calculated the band structure of FeSe, FeSe from which 2 atoms of Se out of 4 per unit cell have been removed and a system in which 2 Te atoms have been substituted for 2 Se atoms in a unit cell in the normal state. In all of the cases the Fermi energies cross multiple bands and there is always at least one conduction band minimum whose energy is lower than the valence band maximum. Hence, on the basis of the calculated band structures, these systems have no semiconducting band gap and should behave as metals. Similarly, there is non-vanishing density of states at the Fermi level. Large effects are found in the density of states. The normal state gap reduces in the composition which becomes superconducting.

The density functional theory is used to optimize the geometry of a cytosine molecule and those of several atoms including H2 molecule, placed in the centre of the hexagon of the cytosine molecule. The adsorption of all of the first group atoms H, Li, Na, K, Cs and Fr, the second group atoms, Be, Mg, Ca, Sr, Ba , the halogens, F, Cl, Br, I and At and the rare gas atoms, He, Ne, Ar, Kr, Xe and Rn has been calculated. All these atoms are optimized and the adsorption energy has been found. The adsorption energy is found to be maximum for Ba, which is a second group element and minimum for hydrogen. In every group, as the atomic number increases, the corresponding adsorption energy increases. Apparently, in several cases, large adsorption energies are found.

We have performed the calculation of structures of clusters containing carbon and nitrogen atoms. We determine the bond lengths in each case. We also calculate the vibrational frequencies of all of the clusters. We compare the calculated values of the vibrational frequencies with those measured by the Raman spectra of amorphous carbon nitrides. Some of the calculated frequencies are in agreement with those measured. We identity that linear structures and hence “back bones” are present in the glassy state.

We have used the density-functional theory to make models containing arsenic and oxygen atoms. The structures are optimized for the minimum energy of the Schrodinger equation. In this way, we obtain the bond distances and angles of the stable structures. We obtain the vibrational frequencies of each cluster. The calculated vibrational frequencies are compared with those found in the experimental Raman spectra. The values of the vibrational frequencies calculated for As O 2 ,As O 4 ( T d ) ,As O 2 (rectangular), As O 2 (triangular) and As O 3 (pyramidal) agree with those found in the Raman spectra of vitreous aresenic oxide, indicating that these clusters are really present in the arsenic oxide glass.

We have used the density functional theory to make the models of Ge x Se 1− x glass for which the energy is a minimum. The clusters, Ge 2 Se 2 , Ge 2 Se 3 , Ge 3 Se, Ge 3 Se 2 , Ge 4 Se, GeSe 3 , GeSe 4 , chain mode zig-zag Ge 4 Se 3 , corner sharing GeSe 4 , and edge sharing Ge 2 Se 6 , have been made successfully and their vibrational spectra have been calculated from the first principles. We are able to optimize the bond distances as well as the bond angles. The calculated values of the frequencies of vibrations of the various clusters have been compared with those obtained from the experimental Raman spectra of actual glasses, Ge x Se 1− x (0 x x within 0.25 nm is nonuniform in the amorphous material. When the same cluster occurs in two stable configurations, low frequency vibrations of frequency, ν −1 , are found. The corner sharing GeSe 4 has low frequency modes at 54 cm −1 and 93 cm −1 whereas these modes disappear in the pyramidal configuration. The low frequency modes are therefore associated with the breaking of C 4 symmetry of the pyramidal configuration. The computed vibrational frequencies of clusters Ge 3 , Ge 4 Se 3 , Ge 2 Se 3 , GeSe 3 and Ge 3 Se 2 are actually present in the Raman spectra of the glass, Ge x Se 1− x (0 x

We have calculated the electronic correlation energy and the exchange energy of a system of GaAs lattice using the density functional theory. In 15 different calculations, we are able to find the energy values for different number of Al and Ga atoms. We have performed the calculation of the energy of the Schroedinger equation in three different approximations, namely, the local density approximation (LDA), the generalized gradient approximation with exchange (GGA-X) and the generalized gradient approximation with exchange and correlations (GGA-XC). The correlation energy has a peak for the cell which has 9 Ga and 5 Al atoms. In this way, we find that the material for which the composition is in the ratio of Ga:Al = 9:5, has the strongest electron correlations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The quantum Hall effect in semiconductor heterostructures is explained by two signs in the angular momentum j=l±s and g=(2j+1)/(2l+1) along with the Landau factor (n+1/2). These modifications in the existing theories explain all of the fractional charges. The helicity which is the sign of the product of the linear momentum with the spin p.s plays an important role for the understanding of the data at high magnetic fields. In particular it is found that particles with positive sign in the spin move in one direction and those with negative sign move in another direction which explains the up and down stream motion of the particles.

The vibrational frequencies of Ge4, S4, I4, SI3, Ge2I2, S2I2, Ge2S2, Ge2SI, GeS3, GeI2S, Ge3S and GeI3 are computed using first principles. These frequencies are compared with those found experimentally in the Raman spectra of Ge0.25S0.75−xIx glass. In this way, it is found that Ge4, GeI2S and GeI3 tetrahedra occur in the glass. These tetrahedra are of different sizes so they stack to form the floppy phase of the glass.

Density functional theory is used to optimize hexagons of all the atoms from the atomic number 1–103. It is found that the hexagon bond length oscillates as one goes along the periodic table. These oscillations can be understood in terms of the electronic structure. A molecule of hydrogen is adsorbed on the surface of group II atoms Be, Mg, Ca, Sr, Ba and Ra. The adsorption energy is the largest for Be and decreases in going from Be to Sr but increases for Ba. The hexagonal bond length increases from 2.24 A for Be to 4.4 A for Ra. Similarly, adsorption of hydrogen molecule on surfaces of rare gas atoms He, Ne, Ar, Kr, Xe and Rn has been optimized. The adsorption energy varies from 0.048 ev for He to 0.025 ev for Rn. Adsorption of hydrogen molecule is also obtained successfully on chalcogens O, S, Se, Te and Po and discussed. Here the adsorption energy is largest for O and smallest for Po. In this way the adsorption of hydrogen molecule on surfaces of 22 different atoms is successfully obtained.

The amplitude of displacement of atoms due to scattering at the surface is calculated. It is found that the zero-point vibrational correction to the mean-square displacement varies as the inverse s...

In two-dimensional electron gas when a large magnetic field is applied in one direction and an electric field perpendicular to it, there is a current in a direction perpendicular to both. This current is called the Hall effect. It remained without quantization until 1980 when it was found that the quantization leads to correct measurement of h/e2. Therefore the quantized Hall effect was further studied at high magnetic fields where fractional quantization was found. The fractional charge can arise from the “incompressibility” in the flux quantization. Laughlin wrote a wave function, the excitations of which are fractionally charged quasiparticles. This wave function comes in competition with charge density waves but for a few fractions it does give the ground state. If “incompressibility” is not considered and it is allowed to be compressible, the fractional charge can arise from the angular momentum which appears in the Bohr magneton in the form of g values. Usually the positive spin is considered but we consider both the positive as well as the negative values so that there is a spin-charge coupling. The values thus calculated for the fractional charge agree with the experimental data on the quantum Hall effect. We have followed this subject for a long time and hence have reviewed the subject. There are several interesting concepts which we learn from this subject. The concept of the Hall effect is quite clear particularly when combined with the flux quantization. We learn about the Landau levels and hence the boson character of electrons in two dimensions. We learn that charge becomes a vector quantity and there is spincharge coupling.

We study the band structure of antiferromagnetic AxFe2Se2 (A = K, Rb) superconductors by using first-principles electronic structure calculations which is density functional theory. In the vicinity of iron-vacancy, we identify the valence electrons of AxFe2Se2 will be filled up to the Fermi level and no semiconducting gap is observed. Hence, the AxFe2Se2 is a metallic instead of semiconducting which leads to superconductivity in the orbital-selective Mott phase. Similarly, there is non-vanishing density of states at the Fermi level.

We find that the phonons of certain frequencies are scattered by the boundary of the nanocrystal so that the specific heat depends on the size of the particles. At low temperatures, the Debye's T3 law, which is known for a large number of particles, is completely broken and exponential behavior may be observed. In the case of a magnetically ordered ferromagnet, we find that the specific heat depends on the size of the nanocrystals because of the scattering of spin waves from the surface of the nanoferromagnets.

We find that in the presence of an electric field there is an attractive intergrain interaction in superconductors which is small. When charge on the ball is permitted to vary with the ball radius, very large balls can be formed. The pairing energy makes the ball compact and hence reduces the size of the ball compared with the classical value. The ball radius depends on the gap of the superconductor due to Josephson tunneling., 8 pages TeX

We have constructed a model of a zeolite which we have optimized using density-functional theory (DFT) in local-density approximation. We have calculated the energy of the zeolite model with and without one potassium atom. From these calculations the adsorption energy of the K atom is found to be −1.057 eV. We find that the bonds in the zeolite ring show an expansion upon addition of a K atom. We have also calculated the spin-dependent bond energy for two K atoms in a zeolite, from which we predict that ferromagnetic order occurs.

The structural distortions, magnetism and superconductivity coexist in single crystals of TmNi2B2C. It is antiferromagnetic below T N ≈1.5 K and superconducting below T c ≈10.6 K . We present here a neutron diffraction study which shows that the apex angle β never reaches the value of 90° belonging to the square lattice because of the onset of a phenomenon that is a precursor to antiferromagnetism. At some field such as 3 kOe upon cooling below 4 K, instead of increasing, the apex angle starts reducing. The angle tends to zero at the Neel temperature and exhibits a mean field exponent of 1/2. Similarly, upon warming above 4 K, the angle becomes soft at the superconducting transition temperature, Tc. The change in London penetration depth as a function of temperature has been deduced from the intensity of the neutron diffraction. This measurement also shows node-like dependence as in the d-wave gap at low temperatures and the s-wave at high temperatures. The nodes are caused by d-wave admixture in the s wave. Therefore, we understand that the gap in the TmNi2B2C has a complex, d+is, symmetry.

We report that the masses of two quasiparticles can be equal only when they are particle–hole conjugates. We suggest several particle–hole conjugate pairs and discuss the role of spin.

Density-functional-theory calculations for the adsorption of molecular nitrogen on ruthenium surface are reported. It is found that nitrogen molecule is adsorbed while standing perpendicular to the triangular surface of ruthenium. The doping by K atoms favours the adsorption of molecular nitrogen. The calculation of the adsorption energy has also been performed on iron and osmium clusters which shows that ruthenium adsorbs better than iron or osmium. It is found that potential for atomic nitrogen is deeper and more stable than that of molecular nitrogen and hence atomic nitrogen is important for catalysis., Comment: MS Word generated HTML, no figures, 5 tables

Flux is known to be quantized in units of hc/2e where h is the Plank's constant, c is the velocity of light and e is the charge of the electron. The quantum theories allow (hc/2e) (n + 1/2) which makes half of the value as the smallest flux quantum. In the case of paramagnetic Meissner effect, we find that 1/4 of the flux quantum occurs. The magnetization shows oscillations as a function of magnetic field where more oscillations are found than is predicted by rational fractions. Therefore, we report that there are fractions of flux other than those given by (hc/2e) (n+1/2). Since φ0 is assumed to be a fundamental constant, the fraction of φ0 indicate that there is a new theory. Thus fractions as well as integer multiples of φ0 occur. In a matrix of wires containing Josephson junctions below a temperature of 0.3 K, the resistivity as a function of flux clearly shows that fractional flux occurs. The values of 1/2, 2/5, 1/3 and 1/4 are clearly seen. We have found that the imaginary part of the susceptibility...

Since the area of a star-shaped vortex is equal to that of a triangular vortex due to the flux quantization, the size of the star-shaped vortex is derivable from that of the triangular vortex. The fluctuation in the distance between vortices is used to find the flux-lattice melting temperature. It is found that the flux-lattice melting temperature of a star-shaped vortex is about one fourth that of a triangular vortex but at zero temperature, the melting field of a star-shaped vortex is about equal to that of the triangular vortex.

We show that the apex angle of the rhombic cell of YNi 2 B 2 C decreases as a function of increasing magnetic field up to a certain constant field, B o ≃1130±70 Oe. This dependence is consistent with the mean field theory of soft vortices. In ErNi 2 B 2 C, the magnetic penetration depth, also exhibits a soft vortex type behaviour below B o ≃250±10 G. The penetration depth is found to have a component which varies as B 1/2 [1−( B / B o )] −1/4 as predicted by the mean field theory and shows divergence at B o . In both the samples, the theoretical predictions are in accord with the experimental measurements. This is the first time that London penetration depth associated with a soft vortex has been reported.

We find that there is a range of current-loop sizes and a range of temperatures under which the para-Meissner effect is predicted. When the phase φ / φ 0 of the Josephson Hamiltonian varies in a certain range, the magnetization becomes positive. In general, the magnetization can be both positive as well as negative with zero resistivity in all phases. The susceptibility as a function of temperature at small magnetic fields is explained on the basis of Josephson interaction. The transition temperature of the para-Meissner effect, T pM , is different from that of the Meissner effect, T c > T pM . The experimental measurements of the magnetization of Tl 2 CaBa 2 Cu 2 O 8 at low fields are in agreement with the theoretical predictions.

The equation of motion of a fluxoid with viscous forces is solved to calculate the effective London penetration depth which is found to vary as the square root of the magnetic field. The London penetration depth as a function of magnetic field in single crystals of is in reasonable agreement with the theory.

Two new processes occur in a superconducting film when it is used as a detector of X-rays. One of these processes is the scattering of the X-ray by a single electron which gives rise to the broadening of the X-ray line. Another process describes the breaking of a Cooper pair by the X-ray which also contributes to the width of the X-ray. The line arising from the single electron process depends on T4 whereas that arising from the pair breaking process varies almost as T6 at low temperatures. Lines occur at ℏωq ± 2Δ, and at ℏωq where ℏωq is the energy of the X-ray and 2Δ is the gap of the superconductor.

We have re-examined the electron - phonon interaction from the viewpoint of its dependence on the crystal structure, particularly on the positioning of atoms in the unit cell. It is found that the strength of the interaction depends on the number of Cu - O layers per unit cell only when there are defects. Accordingly the transition temperature depends on the variable number n of Cu - O layers per unit cell. The larger the number of layers, the larger the transition temperature. We find that the lattice waves can be treated using spherical Bessel functions, and that the electron - phonon interaction oscillates as a function of the number of Cu - O planar layers. The transition temperature of the superconductor then oscillates as a function of n, having a large value at n = 3 and smaller value at n = 4. We predict that the next maximum value of occurs at n = 7. The predicted variation of as a function of n is in reasonable agreement with the experimental values.

The second-order Doppler shift (SODS) of the Mössbauer spectra in a superconductor, δ SC , is calculated for the first time. The shift depends on the fourth power of the ratio of the gap energy of the superconductor to that of the matrix element of the electron–phonon interaction. At T=T c where the gap vanishes, the shift reduces to that in the normal state. The relative values of the SODS and the isomer shifts are found in several superconductors. The isomer shift for 119 Sn in EuBa 2 Cu 2.98 Sn 0.02 O 7−δ has been found. The recoilless fraction of the 119 Sn in EuBa 2 Cu 2.98 Sn 0.02 O 7−δ is compared with a previous theory. It is found that the ordinary BCS type gap agrees with the experimental measurements of the recoilless fraction only when the gap is modified to include a lattice distortion in the superconducting state.

Fluctuation in the distance between flux lines in a superconductor is calculated in a fractal dimension and from that using Lindemann criterion of melting, a field versus temperature phase boundary is deduced for the dendritic phase of a superconductor. This phase boundary is predicted to be B−4∝(1−T FM /T c ) where T FM is the flux-lattice melting temperature. When both the fractal as well as the spectral dimensionalities are considered the phase boundary is predicted to be determined by B8/5∝(1−T FM /T c ).

We propose two mechanisms for the variation of resistance of a film as a function of applied electric field. The resistivity calculated for the field-driven movement of the mobile charge carriers in the normal state of a superconductor in the static approximation varies as the square root of the electric field, ρ∝E1/2 only as long as the dielectric constant is independent of the field. We also predict that the resistivity of a system with conduction electrons localized along a 1-dimensional chain varies as the square root of the applied electric field. We have found that for the electric field larger than about half of the break down field, E >E BD /2, the measured resistivity varies as the square root of the electric field, ρ∝E1/2. Thus both of the theoretical approximations are in agreement with the experimental measurements of the normal state resistivity of YBa 2 Cu 3 O 7−x.

The complete and the correct interpretation of all of the fractions found in the quantum Hall effect of AlGaAs/GaAs heterostructures is presented by use of a specific treatment of spin. Hence the flux quantization depends on spin and the Lande's g value expression is replaced by a formula which exhibits particle-hole symmetry.

We start with the Laughlin's wave function in the complex two-dimensional plane, z=x+iy, and write it in the spinor representation so that the electrons are arranged according to the group theoretical representation so that it is possible to find the ground state energy for a small number of electrons, N=10. We construct another wave function which is appropriate to the quasiparticles of the quantum Hall effect. (QHE). A projection of the QHE wave function on the Laughlin's wave function with appropriate product over all of the quasiparticles produces a new wave function. In this way, we can generate a series of wave functions. We also explain the experimental data of the QHE in graphite.

We find that, in case, the area in which flux is quantized becomes larger than πλ2, it is possible that low fields exist for which the susceptibility is positive. Thus, in the superconductors for which the susceptibility tends to be - unusual positive values are predicted. It is found that the low-temperature positive susceptibility becomes negative at higher temperatures. We calculate the temperature at which the positive susceptibility becomes negative. We also predict that as the magnetic field increases, the positive susceptibility becomes negative. The experimental measurements of magnetization of Bi2Sr2CaCu2O8+σ show that at small fields, the susceptibility does indeed become positive which becomes negative at higher temperatures and at higher fields. As the field increases, the temperature at which the susceptibility becomes zero, reduces until at highfields, where X = 0 point does not occur. A Josephson type free energy gives positive magnetization when the phase angle becomes larger than π. It is consistent with the microwave absorption also.

We find that the magnetization becomes positive for a given number of Josephson junctions in a sample at low magnetic fields. When the phase angle crosses n π (n = integer) radians, the energy of the system changes sign. Accordingly, the magnetization also changes sign when the phase angle becomes 1 2 (2n + 1) π . Thus we predict an unusual positive magnetization in the Meissner region of the (H,T) phase diagram of a superconductor. The experimental measurements of magnetization of Bi2Sr2CaCu2O8 + σ show that it becomes positive at small fields. In some of the samples, depending on the number of Josephson junctions, the susceptibility as a function of temperature becomes a positive constant which is independent of temperature, T, at low temperatures. At large magnetic fields, the usual negative magnetization is found which corresponds to the Meissner value of the susceptibility, χ = − 1 4π .

In two‐dimensional electron gas when a large magnetic field is applied in one direction and an electric field perpendicular to it, there is a current in a direction perpendicular to both. This current is called the Hall effect. It remained without quantization until 1980 when it was found that the quantization leads to the correct value of h/e2. Therefore, the quantized Hall effect was further studied at high magnetic fields where fractional quantization was found. The fractional charge can arise from the “incompressibility” in the flux quantization. Laughlin wrote a wave function, the excitations of which are fractionally charged quasiparticles. This wave function comes in competition with charge density waves but for a few fractions it gives the ground state. If “incompressibility” is not considered and it is allowed to be compressible, the fractional charge can arise from the angular momentum which appears in the Bohr magneton in the form of g values. Usually the positive spin is considered but we cons...

The energy of the free electron gas is calculated in three dimensions. In this calculation an integral over the wave vector space occurs which was solved by Dingle. In the low‐temperature approximation the integral over the Fermi distribution leads to an expression of the type x/sinh x, called Dingle's formula. The spin symmetry is found to modify this formula, which determines the oscillation amplitude of the resistivity as a function of the magnetic field, called the Shubnikov‐de Haas effect. The theory introduces an effective charge, so that the cyclotron frequency becomes fractionalized, resulting in m/v±, which for v± = 1 becomes m, the electron mass. Thus, we have taken into account the spin‐charge fractions, to obtain the correct mass. For example, at a certain magnetic field, 1.5m is found instead of m. The Shubnikov‐de Haas effect uses quantization of Landau levels but not flux quantization. Hence we find that there is a “quantized Shubnikov‐de Haas effect” which measures m/h2. We determine that ...